To the extent that Islam has had “such close ties” to Judaism and Christianity, it has been because these are the three major monotheistic faiths and they (alone among other faiths around them) share a common heritage and some common beliefs. For these reasons, it is not too surprising that they would share some “close ties.” Of course, there has also been a great deal of strain between these faiths, both long ago (the Crusades) and today (groups like ISIS).
Islam is, avowedly, closely connected to the Judeo-Christian tradition. Islam holds that Allah is the same God as Yahweh/Jehovah/God. Islam reveres the same Biblical prophets as Jews and Christians do (and they see Jesus as a prophet). The difference is that they see the other two faiths as having lost their way. They believe that Judaism and Christianity strayed from God’s original message. Because those two faiths had strayed, God revealed his will to Muhammad. He did so in order to get people to return to what he originally wanted. Thus, Islam sees itself as the true interpretation of a faith that it holds in common with Judaism and Christianity.
By contrast, the other faiths with which Islam came into contact early on were pagan faiths. Muslims hate the idea of polytheism and idolatry. Therefore, other faiths that hold to these beliefs and practices would have been repugnant to them. This would have helped emphasize the connections between their faith and the faith of the other monotheistic religions. Therefore, they would have been more likely to have close ties with Judaism and Christianity.
The Muslims saw themselves, Jews, and Christians as “people of the Book” who had something very much in common and very much in contrast to the faiths of other people. This made it more likely that they would have “such close ties.”